Quiz Show

[MoscowFlyer]

Folks, you are on a quiz show. There are 3 doors and behind one of those doors is ?1 million, you have one guess at getting the right door.
You say to the quiz host you think it's the centre door. The quiz host walks up and opens the door on left to show there is nothing behind it. He now says you can stick with your orginal choice or switch to the other.
Do you a) stick or b) switch?

[Bob Singleton]

I'd always go with my first hunch... if I'm wrong, I'm wrong.

If you switch and then find it was the middle door after all, I think you'd feel worse than sticking to your first choice. Anyway, that's what I'd do.

[Jacques]

Sorry Bob you lose - the answer is to switch. If you stay with your original choice, you'll lose two times out of three. If you switch, you'll win two times out of three. By showing you an empty door after your first choice your odds improve as before you had had only a 1 in 3 chance of being right. Odds were 2 to 1 that the money was behind one of the other two doors--now he's showed you which of the other two doors was empty. So based on that switch and win the money.

[Bob Singleton]

That's total bollocks!

*IF* the money is behind the middle of the three doors, and I choose the middle door, the host opening the left hand door and saying I can switch if I want doesn't make the money move around! Same if I'd chosen the "wrong" door! The money is behind the middle door.

Whether I switch or stick makes no difference. From three doors I have a 1 in 3 chance. From two doors a 1 in 2 chance, but switching doesn't actually improve (or decrease) my chances.

Your bookie must see you coming with a glint in his eye if you believe switching would improve your chances of winning!

[Jacques]

Ahhhh!!! But there are 3 doors, A B and C.

If you pick A, and stick with it, there is a 1/3 chance you are right, correct?
That means there is a 2/3 chance that the prize is in either B or C. So if you switch without any extra info, you have to pick between B or C. That means a 1/2 chance of getting it right multiplied by the 2/3 chance that it is in B or C: 1/2 x 2/3 = 1/3. So there is no benefit of switiching without any extra info.

However if you get the extra info - such as eliminating one of the doors, you have the full 2/3 advantage; you dont have to pick between two anymore.

That is why switching is better.

[Bob Singleton]

Stop trying to confuse yourself.

They say that there are "lies, damn lies and statistics"... and it's I'm sure it's possible to prove that black is white using mathematics.

This is just like the tale of the restaurant bill and the "missing" pound.

Go back to school and learn how to use maths properly... the FACT is that IF the money is behind door B and I choose door B, showing me that door A is empty doesn't make door C a better bet!

[Jacques]

Money behind A You guess C Host must show you B--he has no choice. You switch. You win.
Money behind A You guess B Host shows you C--he has no choice. You switch. You win.
Money behind A You guess A Host shows you either B or C at his option. You switch. You lose.

Money behind B You guess C Host must show you A You switch, you win.
Money behind B You guess B Host may show you A or C at his option. You switch, you lose.
Money behind B You guess A Host must show you C You switch, you win.

Money behind C You guess C Host may show you either A or B at his option. You switch, you lose.
Money behind C You guess B Host must show you A You switch, you win.
Money behind C You guess A Host must show you B You switch, you win.

If you switch, you win 2 times out of 3.

[slamdaddy]

I remember my maths teaching telling us about this, and she said that it had been mathematically proven that switching would give you 2/3 chance but sticking would only give you 1/3 chance of finding the money.

To this day I still think she's speaking out of her arse. In this particular scenario you make the assumption that the shows host knows where the money is beforehand, if he doesn't know then I don't think it makes any difference to switch whatsoever, and I can't be bothered to work it out.

[Bob Singleton]

The original post said "There are 3 doors and behind one of those doors is ?1 million, you have one guess at getting the right door".

If you (think you) only have ONE guess, then the host showing you an empty door and offering you a second guess tells me I've chosen the correct door. If I really only had one guess and had chosen the empty door, why not show me immediately that I'm a loser?

Giving me a "second chance" is a psychological ploy and has nothing to do with mathematics.

If the host DOESN'T know where the money is and just happens to chose an empty door first, before giving me a chance to change my mind, from a mathematical point of view, it makes absolutely no difference at all.

Instead of learning probability from a book of puzzles, learn it from a maths book!

[Jacques]

Let's agree to disagree on this one Bob and leave it at that.